{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,7,6]],"date-time":"2023-07-06T04:52:57Z","timestamp":1688619177757},"reference-count":18,"publisher":"AIP Publishing","issue":"1","content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2004,1,1]]},"abstract":"<jats:p>We present a version of the formalism of Dirac based on nonstandard analysis, allowing us to deal with state vectors and operators using the resources of finite-dimensional linear algebra. The space of state vectors is a nonstandard Hilbert space with hyperfinite dimension, which includes all square-integrable functions, together with vectors representing states of definite position or momentum. Every vector is normalizable, even when its norm is infinite. Observables are represented by Hermitian operators, which are always (hyper)bounded and defined on the whole space. The connection with the standard theory is established by postulating the existence of \u201chyper-observables\u201d and nonstandard states. Each observable in the usual sense appears as a kind of standard-scale approximation of some hyper-observable. We show that the probabilistic predictions are consistent with those of the standard theory. Consistency extends to time evolution, in the sense that if an initial nonstandard state is \u201cnear-standard,\u201d then the state after a finite time shall be infinitely near the standard state obtained through the Schr\u00f6dinger equation.<\/jats:p>","DOI":"10.1063\/1.1631394","type":"journal-article","created":{"date-parts":[[2003,12,22]],"date-time":"2003-12-22T16:18:56Z","timestamp":1072109936000},"page":"1-20","update-policy":"http:\/\/dx.doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":3,"title":["The space of state vectors: A hyperfinite approach"],"prefix":"10.1063","volume":"45","author":[{"given":"J.","family":"Almeida","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica, Instituto Superior T\u00e9cnico, 1049-001\u2009Lisboa, Portugal"}]},{"given":"J.","family":"Teixeira","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Instituto Superior T\u00e9cnico, 1049-001\u2009Lisboa, Portugal"}]}],"member":"317","published-online":{"date-parts":[[2003,12,19]]},"reference":[{"key":"2023070601401295500_r1"},{"key":"2023070601401295500_r2"},{"key":"2023070601401295500_r3"},{"key":"2023070601401295500_r4"},{"key":"2023070601401295500_r5"},{"key":"2023070601401295500_r6"},{"key":"2023070601401295500_r7","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1063\/1.522525","article-title":"Application of nonstandard analysis to quantum mechanics","volume":"16","year":"1975","journal-title":"J. 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Phys."}],"container-title":["Journal of Mathematical Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/jmp\/article-pdf\/45\/1\/1\/11110722\/1_1_online.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/jmp\/article-pdf\/45\/1\/1\/11110722\/1_1_online.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,6]],"date-time":"2023-07-06T03:22:08Z","timestamp":1688613728000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/jmp\/article\/45\/1\/1\/387858\/The-space-of-state-vectors-A-hyperfinite-approach"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,12,19]]},"references-count":18,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2004,1,1]]}},"URL":"https:\/\/doi.org\/10.1063\/1.1631394","relation":{},"ISSN":["0022-2488","1089-7658"],"issn-type":[{"value":"0022-2488","type":"print"},{"value":"1089-7658","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2004,1]]},"published":{"date-parts":[[2003,12,19]]}}}